1. Field of the Description
The present description relates, in general, to transferring a spacecraft such as a satellite from one orbit about a celestial body such as the Earth to a second orbit about the same celestial body, and, more particularly, to methods and systems for generating a fuel efficient transfer orbit or transfer trajectory by leveraging multi-body dynamics on the spacecraft during such an orbital transfer.
2. Relevant Background
There are numerous situations in which it is necessary to move or “transfer” a spacecraft from one orbit to another orbit. For example, many spacecraft are first launched into a low Earth orbit (LEO or initial orbit) and then are transferred to a second Earth orbit. In a particular practical application, commercial communication satellites, broadcast satellites, weather satellites, and many other spacecraft begin in LEO or other initial orbit and then are maneuvered through a transfer orbit or transfer trajectory (or geostationary transfer orbit) into a geostationary Earth orbit (GEO or second orbit).
A GEO is a circular orbit at a fixed radius that causes the spacecraft to orbit 35,786 kilometers above the Earth's equator. An object in such an orbit has an orbital period equal to the Earth's rotational period such that the object appears motionless at a fixed position in the sky to ground observers. This is desirable for many satellites so that satellite antennas on the Earth's surface can be pointed to one fixed position in the sky to communicate with the orbiting satellite. In practice, numerous satellites orbit (or “revolve about”) the Earth in this ring above the Equator, with each satellite in a particular slot or position.
For the owners of such satellites and other spacecraft, it is desirable that the orbital transfer be efficient to limit the required energy to affect the move between orbits such as the move from LEO to GEO. If less fuel is required, it may be much less expensive to launch a satellite and place it in orbit about the Earth. Alternatively, a larger or heavier payload may be placed in a particular orbit using the same or a smaller amount of fuel or using a less powerful propulsion system if the transfer orbit is designed to be more efficient or effective in moving a spacecraft between two orbits.
A basic or fundamental orbital transfer, such as the one developed by Walter Hohmann, involves a tangential maneuver to depart an initial orbit and a second tangential maneuver to enter the final orbit. In this regard, a “maneuver” is a deliberate change in velocity of a spacecraft (e.g., by expending fuel to operate a propulsion system) that results in a change of orbit for the spacecraft. This basic transfer considers the gravitational perturbations of a single point-mass such as mass or body about which the spacecraft is orbiting (e.g., the Earth). A spacecraft transfer to or from elliptical orbits that are coplanar uses a very similar process.
Traditionally, orbit transfers do not require the spacecraft to venture beyond the final orbit (e.g., the GEO) during the transfer and, often, not relatively far beyond the initial orbit (e.g., the LEO). For example, the objective of a designer of a transfer orbit (or transfer trajectory) may be to deliver a satellite from the surface of the Earth to a geostationary orbit or GEO. In such cases, a launch vehicle may place the satellite on a transfer orbit or an initial LEO, and the satellite itself, with its propulsion system, will then maneuver to enter into its final orbit. Orbit transfers, of course, may be constructed between almost any two orbits about any celestial body not just between LEO and a GEO.
A Hohmann-type transfer is a useful and even optimal transfer between two orbits when considering only a single point mass such as the Earth (e.g., for co-planar transfers at the Earth with a final orbit to initial orbit ratio of less than 11.94, the Hohmann transfer is optimal). FIG. 1 illustrates a graphic depiction 100 of the Hohmann transfer 120 between two circular orbits about a celestial body 104, with a similar transfer orbit or trajectory being used to affect a spacecraft transfer between two elliptical orbits. A first maneuver at a point or location (labeled “a”) in the initial orbit 110 that involves a change in velocity, Δva, is used to cause the spacecraft 108 to leave the first or initial orbit 110 and move into or along the Hohmann transfer orbit or transfer trajectory 120. A second maneuver at a point or location (labeled “b”) in the transfer orbit 120 that involves another change in velocity, Δvb, is used to place the spacecraft 108 into the second or final orbit 114 (e.g., a GEO or the like) about the celestial body 104. A relatively large amount of fuel has to be expended even in the optimal Hohmann transfer, but owners of satellites or operators of other spacecraft have typically considered this a necessary expense and an already optimized aspect of placing a spacecraft in orbits.
In practice, it is rare for a mission to be concerned only with co-planar transfers such as shown in FIG. 1. Thus, additional considerations are made, when designing an orbit transfer, for the change of other parameters to the orbit. There are many parameters that can be changed; however, orbit inclination is often of most concern with “inclination” referring to the angle between a plane containing an orbit and a reference plane (e.g., with regard to Earth orbits, the reference plane is generally a plane passing through and containing the Equator and as such inclination is an angle measured between an orbit plane and the Equatorial plane). To modify a spacecraft's orbital inclination, a traditional approach is defined with respect to the flight path angle, the initial velocity, and the inclination change desired. The equation to evaluate maneuver size for inclination change may be stated as:
            Δ      ⁢                          ⁢              v                  i          ,          only                      =          2      ⁢              v        initial            ⁢              cos        ⁡                  (                      ϕ                          f              ⁢                                                          ⁢              pa                                )                    ⁢              sin        ⁡                  (                                    Δ              ⁢                                                          ⁢              i                        2                    )                      ,where Δvi,only is the maneuver or change in velocity, vinitial is the spacecraft's present orbiting velocity, and Δi is the change in inclination. In the simple case of a circular orbit, the cosine of the flight path angle, φfpa, is one and, thus, does not impact the calculation. Additionally this maneuver is traditionally designed to occur at the nodal crossing of the orbit.